Chakrabarti, Arunava

Karmakar, S N

Moitra, R K

IACS Institutional Repository

Indian J. Phys. 69A (1), 163 -  168 (1995)IJP A- an international journalCorrelated disorder induced extended states in one dim ensional latticesA ru n av a  C h a k ra b a r ti , 1 S. N. K arm ak ar a n d  R. K. M o itraSaha Institute of Nuclear Physics1/AF, Bidhannagar, Calcutta 700 064, India.A b s t r a c t  : T h e  ro le o f co rre la ted  d iso rder in  p ro d u cin g  ex­ten d ed  e lec tron ic  s ta te s  in qw asiperiodic an d  ap e rio d ic  sys­tem s is d iscussed . T h e  s ta n d a rd  p ic tu re  th a t  h as  em erged  over th e  p a s t  few years is th a t  d im er ty p e  co rre la tio n s a n d  its  g en era lisa tio n  is responsib le  for th e  ex ten d ed  s ta te s  in qu asip erio d ic  co p per-m ean  an d  perio d -d o u b lin g  chains. O n th e  o th e r  h a n d  a new  ty p e  o f co rre la ted  d iso rd e r is found  to  b e  responsib le  for th e  ap p e a ra n ce  o f ex ten d ed  e lec tron ic  s ta te s  in  ap erio d ic  system s like th e  T hue-M o rse  la ttic e . W e an a ly se  b o th  th e  s itu a tio n s  an d  p resen t num erical resu lts  for th e  T h u e-M o rse  chain .K e y w o r d s  : E x ten d ed  s ta te s , d iso rder, q u asip e rio d ic ity  P A C S  n u m b e r s  : 64 .60.A k, 7 1 .2 7 .+ aBy now , th e re  have been  ex tensive  d iscussions in th e  l ite ra tu re  [1-3] on one d im en sio n a l sy stem s lack ing  p erio d ic ity  w hich show ev idence of extended e lec tro n ic  s ta te s . T h e  und erly in g  cause  for th e  a p p e a ra n ce  of extended s ta te s  in  such  cases has been  tra c ed  to  th e  ex istence  o f a  c e rta in  type of sh o rt ran g e  c lu ste rin g  effect am ong  th e  a to m s, firs t p o in te d  o u t by D u n lap  e t  al. [1 ] in  th e ir  s tu d y  of a  d is tr ib u tio n  of ra n d o m  dim ers on  a host la ttic e . I t  w as show n th a t  a t  a  c e rta in  energy  va lue  th e  co m p o site  transfer m a tr ix  for a  d im er offers id en tity  c o n tr ib u tio n  to  th e  fu ll tran sfe r  m atrix , so th a t  a t  th is  energy  th e  e n tire  la ttic e  effectively behaves as an©1995IACSord ered  chain . T h ese  corre la tions have la te r  been  sh o w n  to  p e rs is t  a t  all len g th  scales ju several q u as ip cn o d ic  la ttic e s  like th e  co p p e r  m ea n  d ra in  and  th e  period  doubling  la ttic e  [2 ], lead in g  to  w ho le  h ie ra rc h ie s  of ex ten d ed  s ta te s .T h e  cen tra l fea tu re  in all th e  above eases is th e  e x is te n ce  o f a  c e r ta in  c lu ste r of a tom s in th e  chain a t  cverjf len g th  scale , th e  c lu s te r  itse lf  consisting  of a rep e titio n  of a  c e rta in  su b c lu s tc r  a  fin ite  n u m b e r  of tim es . T h e  electron ic p ro p erties  for th e  com ple te  ch a in  is d e te rm in e d  by  th e  full tra il si er m atrix  which then  has th e  following s tr u c tu r e  :. . . P P P R P R P P I i . . .  \w here R is the. com posite  tran sfe r m a tr ix  h av in g  itse lf  th e  fo rm  M m, where M  is th e  tran sfe r m a tr ix  of a  su b c lu s te r , m  re p e t i t io n s  of w hich generates a  cluster w ith  tran sfe r m a tr ix  R. H ere we a ssu m e  t h a t  th e  chain is being  describ ed  by th e  usual tig h t b in d in g  A n d e rso n  h a m ib o n ia n  w ith  site  energies < a n d  h o pp ing  m a tr ix  e lem en ts t a rra y e d  in  a c c o rd a n c e  with the  u n derly ing  quasifie riod ic ity  of th e  la ttic e . The, a m p litu d e  o f  th e  w avefunction a t  th e  n th  an d  (n  -f 1  ) th  s ites a rc  re la te d  to  th e s e  a t  th e  Oth an d  1 st s ite  by th e  re la tio n  lI 'n =  T 'l 'o , w h ere  is a  co lu m n  v e c to r  consisting of th e  am p litu d es i/>n+i an d  \j>n.T h e  ap p earan ce  of ex ten d ed  s ta te s  in such a  la t t ic e  is d e p e n d e n t  on a) th e  possib ility  of th e  c lu ste r m a tr ix  R  beco m in g  e q u a l to  th e  id e n tity  m atrix  at som e energy  so th a t  a t  th ese  energ ies th e se  c lu s te rs  m ay  b e  effectively d isregarded  an d  b) th e  possib ility  of th e  re m a in d e r  o f  th e  l a t ­tice described  by th e  tran sfe r m atrices  P  fo rm ing  a  p e r io d ic  ch a in . H ere  we shall be concerned w ith  an  analysis of a) only, s ince  in  b o th  cop p er- m ean chain  as well as th e  period  dou b lin g  la t t ic e  th e  o th e r  c o n d itio n  b ) au to m atica lly  o b ta in s  [ see re f .[2 ] for fu r th e r  d e ta ils  ].It tu rn s  o u t th a t  in all eases of in te re s t th e  m a tr ix  M  is a  2  x  2  u n im o d u lar m a tr ix , so th a t  we m ay ap p ly  a  w ell-know n th e o re m  d u e  to  Cayley an d  H am ilton-  Um. 2{X)I  -w here x =  ( l / 2 ) T r M ,  a n d  Um{x) =  s in (m  +  1 ) 0 /  s in  0, w ith  cos 0 =  x. H ere Um(x) is th e  m th  o rd er C hebyshev  p o ly n o m ia l o f th e  seco n d  kind, t o r  values of energy  co rre sp o n d in g  to  th e  ro o ts  o f th e  e q u a tio n  tfm -i(z )  — 0, we effectively have M m p ro p o rtio n a l to  / ,  so t h a t  th eIM Antmno Chakruhnrn, S N  Kam akar andR  K MoilraCorrelated disorder induced extended states etc 165composite transfer m atrix  R  effectively contributes only a phase factor to the full transm ission. Thus if the remaining lattice forms a periodic lattice consisting of the matrices P } and the  energies obtained as solutions of Um- i(z )  =  0 happen to be within the allowed band of this lattice,then the energies autom atically correspond to extended states.While m any of th e  quasipcriodic lattices support extended states w ith the above type of correlated disorder as the  underlying cause, there are other instances of aperiodic chains where the correlated disorder manis- fests itself in an entirely different way. An example of this is th e  Thue- Morsc chain. In this chain there is numerical evidence of extended states [4j although there is no short range dim er-type positional correlations. As we shall see, another kind of clustering effect shows up in this lattice, which cannot be analysed by the  standard  m ethod outlined above and developed in detail in ref.[l,2].A Thue-M orse (TM ) chain may be tfuilt up by starting  w ith two letters A  and B  and using th e  inflation rule A —* A B  and B  —» B A  repeatedly, where the  letters A  and B  may be thought of as representing the atoms A  and B . T he ratio  of the numbers of A and B  atom s in any generation £ is N A( l ) / N g ( £ )  =  1 . We again use the tight-binding one-band ham iltonian to describe this sylem , and for convenience we set the hopping integral equal to unity.As before, the  am plitude of the  wave function a t the  n -th  site is related to th a t a t the  ffrst site by the following m atrix  product..................A fi  >where M n is the  transfer m atrix  for a single atom.In order to unravel the  correlations th a t are responsible for th e  ex­tended states in a Thue-M orse chain, we first recapitulate the  struc­tural peculiarities of this lattice. At the  very basic level, we may re­gard the atom s A  and  B  to  be th e  basic bnilding blocks of the  TM  lattice; a t the  next level the  pairs A B  and B A  may be regarded as the basic s tarting  elements on which the  repeated application of the  Thue-Morse inflation rules generates the whole lattice. We may suc­cessively consider the  pair of quadruplets A B B A  and B A A B , the  pair A B B A B A A B  and B A A B  A B B A , the pair A B B A B A A B B A A B A B B A  *nd B A A B  A B B  A A B B A B A A B  etc. as the starting  blocks for gener- ating the  Thue-M orse chain by applying the  appropriate inflation rules.166 Arunava Chakrabarti, S N  Karmakar andR KM oitrnEvidently, there are no dimer or higher order atom ic clustering in this lattice. It thus becomes necessary to analyse the  products of transfer matrices corresponding to these strings of atom s in order to understand the conditions under which extended states are supported  by the TM lattice.To handle such long products of matrices we resort to the  following device. Let us begin by resolving the 2 x 2  m atrices M n in a basis formed by the 2 x 2 identity m atrix  /  and the three Pauli m atrices a x , a v and a x. We then have the  expressions for M A and M B in the following form:M A =  a AI  +  Pa <7x +  lA<Jy +  f>A<?z 'M b  =  +  P b <*x +  7 b <t v +  f>B<7zw here a A(B) =  6 a (B)  =  ( E  -  t A{B ))/2, P a {B) =  0 a n d  7 / ( B) =  - i .Using this equation repeatedly, we can easily find th e  forms of the longer m atrix products mentioned above. Interestingly, th e  pair of prod­ucts of matrices MaMb Mb M aMb Ma MaM b ■.. and M b M a M aM bMa M b M b M a . . .  each with 2n elements show a surprising regularity having either of the following formsM a M b M b M a M q M a M a M bM b M a M A M g M a M b M b M a  • andM a M b M b M a M b M a M a M b  ■ ■ ■ M b M a M a M b M a M b M b M a • • •=  a n I  +  7„<tv +  Snc t(n even)=  <*nl +  7 > v  +  Vn a*= q„7 + Pncrx +  7n <ry + Sna z(n  odd).=  a nI  -  p na x +  7 na y +  6na zIt follows th a t the m atrix products M a M b M b M a M b M a M a M b  • ■ • and M b M a M a M b M a M b M b M a ■ • ■ can never be m ade equal a t some value of the energy for even values of n  because their expansions differ in two coefficients 7  and 6 . On the o ther hand, for odd  values of n, the above equations show th a t the m atrix  products becom e equal to each other if f)n =  0 . Thus the energy values for which /?„ =  0 (n odd) are the ones for which the composite transfer m atrices for of the strings of atoms A B B A B A A B  • ■ • and B A A B A B B A  ■ ■ ■ offer identicalCorrelated disorder induced extended states etc 167\ W.2uuihibiuAO(•),100 ISO 200 2S0140120100SO,4040200(e)SO 100 ISO 200 2S0100 ISO 200 2S0i .lulu, | lu i u i lulu, | j , luiSO 100 ISO 200 2S0F ig .l  P lo t of iV'rtl2  versus site num ber n for a  TM  sequence w ith 25G sites. Here ca =  - e g  =  0.5 w ith energies m easured in units of hopping m atrix  elem ent t. F igs.(a), (b ), (c) and (d) correspond respectively to energies1.5,0.337965671230,-1.260097834748 and -1.995507972823.co n trib u tio n s to  th e  fu ll tra n s fe r  m a tr ix  for th e  w hole cha in . S ince th e  T hue-M orse  c h a in  m ay  b e  reg a rd ed  as b u ilt  o u t  of th ese  co m p o site  b locks, the fu ll tra n s fe r  m a tr ix  fo r th e  w hole cha in  now  consists  o f a  p ro d u c t  of iden tical u n im o d u la r  2 x 2  m a tric e s  M , each  co rre sp o n d in g  to  a  b lock o f atom s A B B A B A A B  • • ■ o r B A A B A B B A  • ■T h e  recu rs io n  re la tio n s  co n n ec tin g  th e  a,  /?, 7  a n d  6 b e tw een  succes­sive g e n e ra tio n s  m ay  b e  fo u n d  a fte r  som e a lg eb ra , a n d  i t  is s tra ig h fo rw a rd  to o b ta in  th e  a lg eb ra ic  expressions for th e  energy  values o b ta in e d  fro m  the co n d itio n  /3n =  0 . F o r th e  sake of illu s tra tio n  we d isp lay  in  F ig .l  th e  varia tion  o f |V> | 2  w ith  th e  s ite -in d ex  n fo r a  few selected  energ ies. T h e  sig n a tu re  o f th e  u n d e rly in g  T h u e-M o rse  a p e rio d ic ity  is p a r tic u la r ly  a p ­paren t in  F igs. 1 (a ) , 1(c) a n d  1 (d ); all th e  w ave-functions a re  la ttice -lik e . The g en e ra l c h a ra c te r is tic  of th ese  fu n ctio n s is th a t  w ith  in c reas in g  n , am plitudes te n d  to  c lu s te r  a ro u n d  g ro u p s o f s ites  se p a ra te d  by is lan d s  where th e  am p litu d e s  h av e  very  low values, a  fac t w hich w as also p o in te d  out by R y u  e t  al. [5 ].168 Arunava Chakrabarti, S N Karmakar and R A MoitraT h e  sp irit in which th e  p resen t analysis has been  ca rried  o u t  corre­sponds to th e  rea l space rcno rm alisa tio n  g roup  p o in t o f view . H ere  the rcno rm alisa tion  process becom es ev iden t w hen we resolve th e  com posite  m atrices in th e  basis of th e  Pau li m atrices, a  reso lu tio n  w hich clearly reveals th e  rep ea tin g  ch a rac te r  of th e  s trin g s  of m a tr ic e s  in  a lte rn a te  generations.R e f e re n c e s[ 1  ] D . H. D u n lap , H. L. W u and  P. P h illip s , P h y s. R ev . L c tt \  6 5 , 8 8(1990); H. L. W u an d  P. P h illips, P h y s .R e v .L c tt. 6 6 , 1366 '(1991)[ 2  ] S. Sil, S. N. K arm ak ar , R. K. M o itra  an d  A. C h a k ra b a r ti ,  Phys. Rev. B 48 , 4192 (1993); A. C h a k ra b a r ti , S. N. K a rm a k a r  a n d  R.K. M oitra , P hys.R ev .D  ( to  a p p e a r  )[3 ] A. Sanchez, E. M acia and  E. D om inguez-A dam c, P h y s .R ev .B  4 9 , 147 (1994)[4 ] C. S. R yu , G . Y. O h an d  M. H. Lee, P h y s .R e v .B .4 6 , 5162 (1992) [5 ] C. S. R yu , G. Y. Oh an d  M. H. Lee, P h y s .R cv .B  4 8 , 132 (1993)

Correlated Disorder Induced Extended States in One Dimensional Lattices

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Correlated Disorder Induced Extended States in One Dimensional Lattices

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